Functions of Several Variables

A function is a function of several variables if --- that is, if there is more than one
input variable.

For example a function is
a parametrized surface in . Here's a picture of

Or consider a function . Its
graph is a surface in . Here's a picture of the graph of :

Functions of several variables occur in many real world situations
--- in fact, most measurable quantities depend on many factors or
variables. For example, the temperature at a point in space may be a
function of its coordinates . The ideal gas law relates the pressure p, the volume
V, and the temperature T of an ideal gas. I can regard any one of
these three variables as a function of the other two; for example,
writing views p as a
function of V and T.

Example. A function is defined by

Evaluate , , and .

You substitute values into a function of several variables in the
obvious way. For instance, to evaluate , I set , , and in the formula for f:

Likewise,

Definition. For a function :

(a) The domain is the set of all points in
where f is defined.

(b) The image (or the
range) is the set of all outputs of f in .

Remark. I'm following the usual convention:
For functions to
refer to the set of points in where the function is defined as the domain of the function. Thus, for the function
, the
"domain" is the set of points such that , i.e. .

In more advanced courses, a more precise definition of a function requires that the "domain" be
included as part of the function's definition. In that context, what
we're calling the "domain" is referred to as the natural domain (the "biggest possible
set" where the function is defined).

Example. A function of 2 variables is defined
by

Describe the set of points for which f is undefined. What is the domain of f?

is undefined when the denominator is 0:

Thus, f is undefined for points on either of the lines or .

The domain is the set of points which are not on or -- i.e. the points such that .

Example. A function of 2 variables is defined
by

Describe the set of points for which f is undefined. What is the domain of f?

The natural log function is undefined for inputs which are less than
or equal to 0. So
will be undefined if

That is, f is undefined at points where . They are the points lying on or above the
line :

The domain is the set of points below the line .

Example. A function of 2 variables is defined
by

Describe the set of points for which f is undefined. What is the domain of f?

Since the square root of a negative number is undefined, is undefined for

That is, f is undefined at points lying outside the unit circle .

The domain is the set of points on or inside the unit circle .

Example. Describe the image (or the range) of
the function .

The image of a function is the set of outputs. What are the outputs
of the inverse tangent function ?

For any x and y,

I know that attains every
value between and , and if I set I have .

Therefore, the image of f is .

A function of several variables can be pictured in many ways. For
example, you can draw the graph of a function
of 2 variables . Because
plotting points in 3 dimensions is tedious and difficult, you'd
probably use software to draw the graph.

for functions , you
may also get a "picture" of the function by drawing its level sets. A level set for f is obtained by
setting , where c is a constant. By using
different values for c, you get a picture of the "levels"
of the function.

You may have seen topographic maps, where the
level curves are referred to as contour lines.
They represent lines along which the altitude ("z") is
constant:

Example. Sketch the graph of , and some of the contour lines.

Here's the graph of the function, produced by a computer:

I can use a computer to sketch the level curves, but this example is
simple enough that I'll analyze it first. I get level curves by
setting z to specific numbers and graphing the curves I get.

You can see the level curves are a family of parabolas.

For a function of 3 variables , setting for various numbers c produces level surfaces. If you interpret as the temperature at a point in space, then a level surface is the set of points in space where the
temperature is c.